What are the most common deep reinforcement learning algorithms and models apart from DQN?
There are several common deep reinforcement algorithms and models apart from deep Q networks (or deep Q learning). I will list some of them below (along with a link to the paper that introduced them), but note that some of these may not be state-of-the-art (at least, not anymore, and it's likely that all of these will be replaced in the future).

Double DQN (DDQN) (2015)
Duelling DQN (2015)
Trust Region Policy Optimization (TRPO) (2015)
Deep Deterministic Policy Gradient (DDPG) (2016)
Asynchronous Advantage Actor-Critic (A3C) (2016)
Hindsight Experience Replay (HER) (2017)
Proximal policy optimization (PPO) (2017)
Twin Delayed Deep Deterministic policy gradient algorithm (TD3) (2018)
Soft Actor-Critic (SAC) (2018)

For an exhaustive overview of deep RL algorithms and models, maybe take a look at this pre-print Deep Reinforcement Learning (2018) by Yuxi Li.